How can I find a mesh's position when affected by a constraint?
Presuming you can see the model in object mode, Select it and press SH-S, choose "Cursor to selected".  The 3D cursor will jump to the model's origin point.  Press N and you can read off the 3D cursor's location.

If you want to store this within the Blender file, ADD an Empty.  It will appear at that location and it's own location readings at right will be the same as the 3D cursor's.

You haven't stated what constraint(s) you're inquiring about so I've just tested this out both with a path constraint and parenting the object object to another.  Both work as described.